Question

Find the value of $2\cos \frac{\pi }{13}\cos \frac{9\pi }{13}+\cos \frac{3\pi }{13}+\cos \frac{5\pi }{13}$.

Answer 1

$LHS=2\cos \frac{\pi }{13}\cos \frac{9\pi }{13}+\cos \frac{3\pi }{13}+\cos \frac{5\pi }{13}$

$=2\cos \frac{\pi }{13}\cos \frac{9\pi }{13}+2\cos \frac{(5+3)\pi }{13\times 2}\cos \frac{(5-3)\pi }{2\times 13}$

$=2\cos \frac{\pi }{13}\cos \frac{9\pi }{13}+2\cos \frac{4\pi }{3}\cos \frac{\pi }{13}$

$=2\cos \frac{\pi }{13}(\cos \frac{9\pi }{13}+\cos \frac{4\pi }{13})$

$=2\cos \frac{\pi }{13}\left(2\cos \left(\frac{13\pi }{13\times 2}\right)\cos \left(\frac{5\pi }{13\times 2}\right)\right)$

$=4\cos \frac{\pi }{13}\cos \frac{\pi }{2}\cos \frac{5\pi }{26}$

$=0$ $\because (\cos \frac{\pi }{2}=0)$

$=RHS$